Finite Group Databasebeta

← #223239

Subgroups — #223239

Order 512.

822 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C211yesmaximal, characteristic
2#22797C211yesmaximal, characteristic
3#22749C211yesmaximal, characteristic
4#9907C211yesmaximal, characteristic, complemented
5#9904C211yesmaximal, characteristic, complemented
6#9140C211yesmaximal, characteristic, complemented
7#9137C211yesmaximal, characteristic, complemented
8#9132C211yesmaximal, characteristic, complemented
9#9130C211yesmaximal, characteristic, complemented
10#8726C211yesmaximal, characteristic, complemented
11#8724C211yesmaximal, characteristic, complemented
12#8589C211yesmaximal, characteristic, complemented
13#8233C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic, complemented
15#8043C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20#2859C2211yescharacteristic
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#2570—21no—
24#2511—21no—
25#2507C412yes—
26#2497C412yes—
27#2487C412yes—
28#2442—21no—
29#2441—21no—
30#2423—21no—
31#2417—21no—
32#2360—21no—
33#2352—21no—
34#2345C412yes—
35C4×C4⋊Q8C2211yescharacteristic
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1955—21no—
40#1924—21no—
41#1924—21no—
42#1923—21no—
43#1916—21no—
44#1870—21no—
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1771—21no—
52#1767C2211yescharacteristic
53#1767C2211yescharacteristic
54#1753—21no—
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C22×M16—21no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×#125—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×#117—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2×C412yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#432—22no—
87#432—41no—
88#431—22no—
89#431—22no—
90#431—41no—
91#431—41no—
92#423—22no—
93#423—41no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117#400—22no—
118#400—22no—
119#399—22no—
120#398—22no—
121#398—22no—
122#396—22no—
123#391—21no—
124#391—21no—
125#391—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—22no—
132#389C2×C412yes—
133#389C2×C412yes—
134#389C2×C412yes—
135#389C2×C412yes—
136#389C2×C412yes—
137#388—22no—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#386—22no—
143#385—22no—
144#384—21no—
145#384—21no—
146#384C2×C412yes—
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#382—21no—
151#382—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155#380—22no—
156#380C2×C412yes—
157#380C2×C412yes—
158#380C2×C412yes—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4C2×C412yes—
162C4×C4⋊C4C2×C412yes—
163C4×C4⋊C4C2×C411yescharacteristic
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C22⋊C4—22no—
167C4×C22⋊C4C2×C411yescharacteristic
168C4×C22⋊C4C2×C411yescharacteristic
169C4×C22⋊C4D411yescharacteristic
170C4×C22⋊C4D411yescharacteristic
171C4×C22⋊C4D411yescharacteristic
172C4×C22⋊C4D411yescharacteristic
173C43C2311yescharacteristic
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#332—22no—
183#332—22no—
184#332—41no—
185#332—41no—
186C22×Q8—21no—
187C22×D4C22⋊C411yescharacteristic
188C23×C4—22no—
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—81no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210#125—81no—
211#125—81no—
212C22⋊Q8—42no—
213C22⋊Q8—42no—
214C22⋊Q8—81no—
215C22⋊Q8—81no—
216C4×Q8—81no—
217C4×Q8—81no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—42no—
223#117—81no—
224#117—81no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—81no—
242C2×C4⋊C4—81no—
243C2×C4⋊C4—81no—
244C2×C4⋊C4—81no—
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C22⋊C411yescharacteristic
257C2×C22⋊C4C22⋊C411yescharacteristic
258C2×C22⋊C4C22⋊C411yescharacteristic
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2411yescharacteristic
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300C4⋊C8—81no—
301C4⋊C8#4112yes—
302C4⋊C8#4112yes—
303C22⋊C8—42no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307C22⋊C8—81no—
308C22⋊C8—81no—
309C22⋊C8—81no—
310#97—42no—
311#97—42no—
312#97—81no—
313C4×C8—42no—
314C4×C8—42no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95C22⋊C411yescharacteristic
326#95C22⋊C411yescharacteristic
327#95C22⋊C411yescharacteristic
328#95C22⋊C411yescharacteristic
329#95C22⋊C411yescharacteristic
330#95C22⋊C411yescharacteristic
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×D4—22no—
340C2×D4—41no—
341C2×D4—41no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—44no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4#13712yes—
390C22×C4#13712yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394M16—84no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—44no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—44no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—41no—
479C42—41no—
480C42—82no—
481C42—82no—
482C42C2×#4111yescharacteristic
483C42C2×#4111yescharacteristic
484C42H(5,𝔽2)11yescharacteristic
485C422−1+411yescharacteristic
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23#41912yes—
500C23C22×C22⋊C411yescharacteristic
501Q8—81no—
502Q8—81no—
503Q8—81no—
504Q8—81no—
505D4—44no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—164no—
569C2×C4#41312yes—
570C2×C4#41912yes—
571C2×C4C4×#4111yescharacteristic
572C2×C4#52011yescharacteristic
573C2×C4#54611yescharacteristic
574C2×C4#54611yescharacteristic
575C2×C4#54711yescharacteristic
576C2×C4#54711yescharacteristic
577C2×C4C2×#13711yescharacteristic
578C2×C4C2×#13711yescharacteristic
579C2×C4#57511yescharacteristic
580C2×C4#57511yescharacteristic
581C2×C4#57611yescharacteristic
582C2×C4#57611yescharacteristic
583C2×C4#57711yescharacteristic
584C2×C4#57711yescharacteristic
585C8—42no—
586C8—42no—
587C8—81no—
588C8—81no—
589C8—81no—
590C8—81no—
591C8—81no—
592C8—81no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22#233911yescharacteristic
608C22C2×#41311yescharacteristic
609C22#288911yescharacteristic
610C22C2×#41911yescharacteristic
611C22#289611yescharacteristic
612C22#289711yescharacteristic
613C22#289711yescharacteristic
614C4—21no—
615C4—21no—
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4#330711yescharacteristic
631C4#330911yescharacteristic
632C4#331411yescharacteristic
633C4#331611yescharacteristic
634C4#334411yescharacteristic
635C4#334611yescharacteristic
636C4#334711yescharacteristic
637C4#335011yescharacteristic
638C2—42no—
639C2—81no—
640C2#1774711yescharacteristic
641C2#1776111yescharacteristic
642C2#1776911yescharacteristic
643C2#1777611yescharacteristic
644C2#2248011yescharacteristic
645C2#2249711yescharacteristic
646C2#2249811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9907C2semidirect
5. #9904C2semidirect
6. #9140C2semidirect
7. #9137C2semidirect
8. #9132C2semidirect
9. #9130C2semidirect
10. #8726C2semidirect
11. #8724C2semidirect
12. #8589C2semidirect
13. #8233C2semidirect
14. #8046C2semidirect
15. #8043C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#361892
#227972
#227492
#99072
#99042
#91402
#91372
#91322
#91302
#87262
#87242
#85892
#82332
#80462
#80432